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Unformatted text preview: ent from α , which is the ball’s angular acceleration about
its own center of mass.
If we again make the approximation that sin(θ ) ≈ θ and compare our result to Equation 1, we
obtain
g
ω=
(15)
(1 + I /mr2 ) (R − r)
as the angular frequency of the ball’s oscillation. Recall from Laboratory 6 that
fI ≡ I
mr2 is a unitless number between zero and one called the geometric factor.
Recall, however, that the result in equation (15) was attained by an approximation. The approximate result is very close to exact in the limit that the ball’s radius is much smaller than the dish’s
radius of curvature. We can get an exact expression for ω by using a slightly different approach. It
turns out that we need to make two simple modiﬁcations to our above formulation.
First, we replace Newton’s second law for forces in Equation 9 with Newton’s second law for
torques,
τ net = τ f + τgravity ,
(16)
which states the net torque on the ball equals the sum of the torques, i.e. the torque generated
by friction plus the torque generated by gravity. It is important to recall that in order to calculate
torque, you must specify a pivot point in order to determine the lever arm of the applied force. In
theory, you can choose any point you wish as the pivot point. However, for harmonic motion, there
is usually a particular choice in pivot point that proves much more useful than any other point.
For the simple pendulum, the most useful pivot point is the location where the rod is ﬁxed to the
ceiling. For the ballindish apparatus, the pivot point is the center of curvature of the dish. Using
the deﬁnition for torques introduced in Equation 1 of lab 8, we can rewrite Equation 16 as
Fnet (R − r) = Ff R − Fg (R − r) sin θ , (17) where Fnet = m at , the net force acting on the ball from Equation 9, Fg = m g, the force of gravity,
and Ff is the friction force (see Figure 7). Notice that Equation 17 is just Equation 9 where,
instead, each term is multiplied by its respective leve...
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 Spring '11
 Gerson

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